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If you're saying you have a defective subwoofer that keeps the system from coming up you might find one on eBay. many owners of HTIB's sell off the remaining bits as parts die off.

Or maybe your subwoofer comment is not related to your power problem.

Generally speaking, an amp protects itself from heat, shorts, overloads and operator exuberance by refusing to turn on or stay on.
Overloads can be from excessive periods of high output or marginally low impedance loading by the speakers; and shorts would be wiring issues or a speaker blowing up.

You should be able to feel if it's hot. WHY is it overheating? Make sure it has sufficient ventilation on all sides and that vent holes are not blocked by dust balls. Ensure the fan (if equipped) is running as designed (some only operate on demand). Clean dust and debris from it.

If the amp comes back on after cooling, you're lucky. They only have so many self-protection cycles in their lives so continuously resetting or cycling their power without addressing the cause can do more harm than good.

If it protects immediately on a cool power up you should disconnect the speaker connections and try it 'naked'. If it comes up then diagnose which lead(s) are shorted. If it does not come up the problem is internal and should be left to an experienced and competent hands-on tech.

Check for loose speaker connections at the speaker as a root cause for intermittent shutdown.

A) Amplified Subwoofer. (With 110volts A.C. cord attached to the sub)
The subwoofer electronics circuit needs repair or change.
You will find a compatible subwoofer for this type of subwoofer in audio appliance stores.

B)Not amplified Subwoofer (No 110volts A.C. cord attached to the sub)

Bring your sound system to a technician to have the soldering remaked to the devices bolted on the main heatsink.
And have the Subwoofer checked too.

This type of subwoofer can mostly be buy in kit with the satellites speakers.
And need to match thee impedance of the output of your system.

there will be a subwoofer pre out( needs to be amplified) or if it don't have subwoofer pre out you will have to send the audio out to a stereo reciever that has inputs for the optical ( digital audio signal )( to a stereo or surround system that has 5.1 or higher has full surround sound ) and of course L and R audio inputs and the subwoofer pre out , then the subwoofer pre out will have to be amplified by a reciever , subwoofer amp , surround sound , ect

You may have a break in the wire. You can find out by slowing moving your fingers down the wire until you notice sound coming from the subwoofer. You can fix it yourself by cutting about an inch away from the break both ways. Then cuttings some of the wire off, so the copperwire is exposed. then wrap the two copperwire ends together and seal it with elections tape. or Try replacing the 60 watt subwoofer wire.